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Theorem mircgrextend 26170
Description: Link congruence over a pair of mirror points. cf tgcgrextend 25973. (Contributed by Thierry Arnoux, 4-Oct-2020.)
Hypotheses
Ref Expression
mirval.p 𝑃 = (Base‘𝐺)
mirval.d = (dist‘𝐺)
mirval.i 𝐼 = (Itv‘𝐺)
mirval.l 𝐿 = (LineG‘𝐺)
mirval.s 𝑆 = (pInvG‘𝐺)
mirval.g (𝜑𝐺 ∈ TarskiG)
mirtrcgr.e = (cgrG‘𝐺)
mirtrcgr.m 𝑀 = (𝑆𝐵)
mirtrcgr.n 𝑁 = (𝑆𝑌)
mirtrcgr.a (𝜑𝐴𝑃)
mirtrcgr.b (𝜑𝐵𝑃)
mirtrcgr.x (𝜑𝑋𝑃)
mirtrcgr.y (𝜑𝑌𝑃)
mircgrextend.1 (𝜑 → (𝐴 𝐵) = (𝑋 𝑌))
Assertion
Ref Expression
mircgrextend (𝜑 → (𝐴 (𝑀𝐴)) = (𝑋 (𝑁𝑋)))

Proof of Theorem mircgrextend
StepHypRef Expression
1 mirval.p . 2 𝑃 = (Base‘𝐺)
2 mirval.d . 2 = (dist‘𝐺)
3 mirval.i . 2 𝐼 = (Itv‘𝐺)
4 mirval.g . 2 (𝜑𝐺 ∈ TarskiG)
5 mirtrcgr.a . 2 (𝜑𝐴𝑃)
6 mirtrcgr.b . 2 (𝜑𝐵𝑃)
7 mirval.l . . 3 𝐿 = (LineG‘𝐺)
8 mirval.s . . 3 𝑆 = (pInvG‘𝐺)
9 mirtrcgr.m . . 3 𝑀 = (𝑆𝐵)
101, 2, 3, 7, 8, 4, 6, 9, 5mircl 26149 . 2 (𝜑 → (𝑀𝐴) ∈ 𝑃)
11 mirtrcgr.x . 2 (𝜑𝑋𝑃)
12 mirtrcgr.y . 2 (𝜑𝑌𝑃)
13 mirtrcgr.n . . 3 𝑁 = (𝑆𝑌)
141, 2, 3, 7, 8, 4, 12, 13, 11mircl 26149 . 2 (𝜑 → (𝑁𝑋) ∈ 𝑃)
151, 2, 3, 7, 8, 4, 6, 9, 5mirbtwn 26146 . . 3 (𝜑𝐵 ∈ ((𝑀𝐴)𝐼𝐴))
161, 2, 3, 4, 10, 6, 5, 15tgbtwncom 25976 . 2 (𝜑𝐵 ∈ (𝐴𝐼(𝑀𝐴)))
171, 2, 3, 7, 8, 4, 12, 13, 11mirbtwn 26146 . . 3 (𝜑𝑌 ∈ ((𝑁𝑋)𝐼𝑋))
181, 2, 3, 4, 14, 12, 11, 17tgbtwncom 25976 . 2 (𝜑𝑌 ∈ (𝑋𝐼(𝑁𝑋)))
19 mircgrextend.1 . 2 (𝜑 → (𝐴 𝐵) = (𝑋 𝑌))
201, 2, 3, 4, 5, 6, 11, 12, 19tgcgrcomlr 25968 . . 3 (𝜑 → (𝐵 𝐴) = (𝑌 𝑋))
211, 2, 3, 7, 8, 4, 6, 9, 5mircgr 26145 . . 3 (𝜑 → (𝐵 (𝑀𝐴)) = (𝐵 𝐴))
221, 2, 3, 7, 8, 4, 12, 13, 11mircgr 26145 . . 3 (𝜑 → (𝑌 (𝑁𝑋)) = (𝑌 𝑋))
2320, 21, 223eqtr4d 2824 . 2 (𝜑 → (𝐵 (𝑀𝐴)) = (𝑌 (𝑁𝑋)))
241, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23tgcgrextend 25973 1 (𝜑 → (𝐴 (𝑀𝐴)) = (𝑋 (𝑁𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2050  cfv 6188  (class class class)co 6976  Basecbs 16339  distcds 16430  TarskiGcstrkg 25918  Itvcitv 25924  LineGclng 25925  cgrGccgrg 25998  pInvGcmir 26140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-trkgc 25936  df-trkgb 25937  df-trkgcb 25938  df-trkg 25941  df-mir 26141
This theorem is referenced by:  mirtrcgr  26171
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